Question

A box with a square base and closed top is required to have a volume of 27,000 cm3
. Find the dimensions of the box that minimize the amount of material used.

Answer 1

Answer:

h=30cm

Base sides, x=30cm

Step-by-step explanation:

Let x be the dimension of the sides and h the dimension of height;

$V=x^2h$

#Material used is directly proportional to surface area. Minimizing surface area will minimize the amount of material used.

$Area=2x^2+4xh$

Find area as a function of x:

$V=x^2h=27000\\\\h=\frac{27000}{x^2}\\\\Area=2x^2+4x(\frac{27000}{x^2})\\\\=2x^2+\frac{108000}{x}$

To minimize Area, we find the first derivative of Area:

$A=2x^2+\frac{108000}{x}\\\\A\prime=4x+\frac{108000}{x^2}=0\\\\\frac{4x^3-108000}{x^2}=0\\\\4x^3=108000\\\\x=30$

#find the second derivative to verify the minimum:

$A\prime=4x+\frac{108000}{x^2}\\A\prime\prime=4x+\frac{108000}{x^3}=124, x=30$

Substitute x in the volume equation to find h:

$h=\frac{27000}{x^2}\\\\=\frac{27000}{30^2}\\\\h=30$

Hence, the dimensions of the box that minimize the amount of material used h=30cm and x=30cm